Nuclear Physics B258 (1985) 157-178 © North-Holland Publishing Company ABELIAN GAUGE STRUCTURE INSIDE NONABELIAN GAUGE THEORIES Yong-Shi WU ~ and A. ZEE Department of Physics, FM-15, University of Washington, Seattle, WA 98195, USA Received 4 February 1985 We show that the inclusion of topological lagrangians in nonabelian gauge theories introduces certain topologically nontrivial abelian background fields in the configuration space of these theories. In particular, the 0-term and the topological mass term lead, respectively, to a vortex and a monopole in gauge orbit space in 3 + 1 and 2 + 1 dimensions. The identification of the theta-vacuum effect with a kind of Bohm-Aharonov effect in gauge-orbit space motivates a discussion on the possibility of solving' the strong CP problem dynamically. 1. Introduction In this paper we will point out that nonabelian gauge theories with topological terms included contain within them a hidden abelian structure. The point is that nonabelian gauge theories, when quantized in the Schr6dinger formulation, may be described as the quantum mechanics of a point particle moving in a background abelian gauge field. We study in detail the (3 + D-dimensional Yang-Mills theory with the Pontryagin term included [1] and the (2 + 1)-dimensional Yang-Mills theory with the Chern-Simons term included [2]. We determine the nature of the abelian gauge fields in these two cases. Interestingly enough, we find the field of a vortex in the (3 + 1)-dimensional case, and the field ofa monopole in the (2 + 1)-dimensional case. (Note that there is a curious reversal here since, normally, a monopole resides naturally in 3-dimensional space while a vortex resides naturally in 2-dimensional space.) The identification of the hidden abelian gauge field as that of a vortex allows us to speculate on a possible solution of the strong CP problem. There are, however, considerable obstacles to realizing this solution. We will discuss the proposed solution and the obstacles in detail. This paper will be organized as follows. In sect. 2 we show how to introduce an abelian gauge structure within gauge theories with topological terms included. In sect. 3 we present a homotopic analysis of the gauge-orbit space which is helpful to subsequent sections. Sect. 4 is devoted to the study of the (3+l)-dimensional Yang-Mills theories with the Pontryagin term included. On the basis of this study, we speculate, in sect. 5, on a possible solution to the strong CP problem and discuss i On leave from the current address: Department of Physics, University of Utah, Salt Lake City, UT 84112, USA. 157